In this note, we consider Szemerédi’s theorem on k-term arithmetic progressions over finite fields \(\mathbb {F}_p^n\) , where the allowed set S of common differences in these progressions is chosen randomly of fixed size. Combining a generalization of an argument of Altman with Moshkovitz–Zhu’s bounds for the partition rank of a tensor in terms of its analytic rank, we (slightly) improve the best known lower bounds (due to Briët) on the size |S| required for Szemerédi’s theorem with difference in S to hold asymptotically almost surely.
